Finite Group Databasebeta

← #2076415

Subgroups — #2076415

Order 512.

3000 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16429C213yesmaximal, complemented
2#16318C213yesmaximal, complemented
3#16005C213yesmaximal, complemented
4#15823C213yesmaximal, complemented
5#15622C213yesmaximal, complemented
6#15544C213yesmaximal, complemented
7#15467C211yesmaximal, characteristic
8#15297C213yesmaximal
9#14693C213yesmaximal
10#14547C213yesmaximal, complemented
11#14433C213yesmaximal, complemented
12C25×C4C2211yescharacteristic
13C2×#397C2213yes—
14C2×#396C2213yes—
15C2×#395C2211yescharacteristic, complemented
16C2×#395C2213yescomplemented
17C2×#393C2213yes—
18C2×#388C2213yes—
19C2×#387C2213yescomplemented
20C2×#385C2213yes—
21C2×#385C2213yes—
22C2×#385C2213yes—
23C2×#385C2213yes—
24C2×#385C2213yes—
25C2×#385C2213yes—
26C2×#383C2211yescharacteristic, complemented
27C2×#379C2213yes—
28C2×#379C2213yes—
29#1445C2213yescomplemented
30#1445C2213yescomplemented
31#1445C2213yescomplemented
32#1444C2211yescharacteristic, complemented
33#1443C2213yescomplemented
34#1443C2213yescomplemented
35#1443C2211yescharacteristic, complemented
36#1442C2213yescomplemented
37#1442C2213yescomplemented
38#1442C2213yescomplemented
39#1442C2213yescomplemented
40#1442C2213yescomplemented
41#1441C2213yescomplemented
42#1441C2213yescomplemented
43#1441C2213yescomplemented
44#1441C2213yescomplemented
45#1440C2213yes—
46#1440C2213yes—
47#1440C2213yes—
48#1440C2213yes—
49#1440C2213yes—
50#1439C2213yes—
51#1439C2213yes—
52#1439C2213yes—
53#1439C2213yes—
54#1437C2213yes—
55#1437C2213yes—
56#1437C2213yes—
57#1436C2213yescomplemented
58C4⋊#95C2213yescomplemented
59C4⋊#95C2213yescomplemented
60#1434C2213yescomplemented
61C22⋊#95C2213yes—
62C22⋊#95C2213yes—
63C22⋊#95C2213yes—
64C22⋊#95C2213yes—
65C22⋊#95C2213yes—
66C22⋊#95C2213yes—
67C26C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C24×C4—46no—
70C24×C4—42no—
71C24×C4—46no—
72C24×C4C2313yes—
73C24×C4C2313yes—
74C22×C4⋊C4C2313yes—
75C22×C4⋊C4C2313yes—
76C22×C4⋊C4C2313yes—
77C22×C4⋊C4C2313yes—
78C22×C4⋊C4C2313yes—
79C22×C4⋊C4C2313yes—
80C22×C4⋊C4C2313yes—
81C22×C22⋊C4C2313yes—
82C22×C22⋊C4C2313yes—
83C22×C22⋊C4C2313yes—
84C22×C22⋊C4C2313yes—
85C22×C22⋊C4C2313yes—
86C22×C22⋊C4C2313yes—
87C22×C22⋊C4C2313yes—
88C22×C22⋊C4C2313yes—
89C22×C22⋊C4C2313yes—
90C22×C22⋊C4C2313yes—
91C22×C22⋊C4C2313yes—
92C22×C22⋊C4C2313yes—
93C22×C42C2313yes—
94C22×C42C2311yescharacteristic
95#397—46no—
96#396—212no—
97#395—42no—
98#395—46no—
99#393—46no—
100#388—212no—
101#387—46no—
102#385—46no—
103#385—46no—
104#385—46no—
105#385—46no—
106#385—46no—
107#385—46no—
108#383—42no—
109#379—212no—
110#379—46no—
111C2×#95C2313yes—
112C2×#95C2313yes—
113C2×#95C2313yes—
114C2×#95C2313yes—
115C2×#95C2313yes—
116C2×#95C2313yes—
117C2×#95C2313yes—
118C2×#95C2313yes—
119C2×#95C2313yes—
120C2×#95C2313yes—
121C2×#95C2313yes—
122C2×#95C2313yes—
123C2×#95C2313yes—
124C2×#95C2313yes—
125C2×#95C2313yes—
126C2×#95C2313yes—
127C2×#95C2313yes—
128C2×#95C2313yes—
129C2×#95C2313yes—
130C2×#95C2313yes—
131C2×#95C2313yes—
132C2×#95C2313yes—
133C2×#95C2313yes—
134C2×#95C2313yes—
135C2×#95C2313yes—
136C2×#95C2311yescharacteristic
137C2×#95C2313yes—
138C2×#95C2313yes—
139C2×#95C2313yes—
140C2×#95C2311yescharacteristic
141C25—26no—
142C25—26no—
143C25—26no—
144C25—43no—
145C25—41no—
146C25—41no—
147C25—41no—
148C25C2413yes—
149C22×Q8—43no—
150C22×Q8—43no—
151C22×Q8—41no—
152C23×C4—26no—
153C23×C4—26no—
154C23×C4—26no—
155C23×C4—26no—
156C23×C4—26no—
157C23×C4—26no—
158C23×C4—43no—
159C23×C4—43no—
160C23×C4—43no—
161C23×C4—43no—
162C23×C4—43no—
163C23×C4—43no—
164C23×C4—43no—
165C23×C4—43no—
166C23×C4—416no—
167C23×C4—448no—
168C23×C4—448no—
169C23×C4C2413yes—
170C23×C4C2413yes—
171C23×C4C2413yes—
172C23×C4C2413yes—
173C23×C4C2413yes—
174C23×C4C2413yes—
175C23×C4C2413yes—
176C23×C4C2413yes—
177C23×C4C2413yes—
178C23×C4C2411yescharacteristic
179C2×C4⋊C4—26no—
180C2×C4⋊C4—26no—
181C2×C4⋊C4—26no—
182C2×C4⋊C4—26no—
183C2×C4⋊C4—26no—
184C2×C4⋊C4—26no—
185C2×C4⋊C4—26no—
186C2×C4⋊C4—26no—
187C2×C4⋊C4—43no—
188C2×C4⋊C4—43no—
189C2×C4⋊C4—43no—
190C2×C4⋊C4—43no—
191C2×C4⋊C4—43no—
192C2×C4⋊C4—43no—
193C2×C4⋊C4—43no—
194C2×C4⋊C4—43no—
195C2×C4⋊C4—43no—
196C2×C4⋊C4—43no—
197C2×C4⋊C4—43no—
198C2×C4⋊C4—43no—
199C2×C4⋊C4—43no—
200C2×C22⋊C4—26no—
201C2×C22⋊C4—26no—
202C2×C22⋊C4—26no—
203C2×C22⋊C4—26no—
204C2×C22⋊C4—26no—
205C2×C22⋊C4—26no—
206C2×C22⋊C4—26no—
207C2×C22⋊C4—26no—
208C2×C22⋊C4—26no—
209C2×C22⋊C4—26no—
210C2×C22⋊C4—26no—
211C2×C22⋊C4—26no—
212C2×C22⋊C4—26no—
213C2×C22⋊C4—26no—
214C2×C22⋊C4—26no—
215C2×C22⋊C4—26no—
216C2×C22⋊C4—26no—
217C2×C22⋊C4—26no—
218C2×C22⋊C4—26no—
219C2×C22⋊C4—43no—
220C2×C22⋊C4—43no—
221C2×C22⋊C4—43no—
222C2×C22⋊C4—43no—
223C2×C22⋊C4—43no—
224C2×C22⋊C4—43no—
225C2×C22⋊C4—43no—
226C2×C22⋊C4—43no—
227C2×C22⋊C4—43no—
228C2×C22⋊C4—43no—
229C2×C22⋊C4—43no—
230C2×C22⋊C4—43no—
231C2×C22⋊C4—43no—
232C2×C22⋊C4—43no—
233C2×C22⋊C4—43no—
234C2×C22⋊C4—43no—
235C2×C22⋊C4#41112yes—
236C2×C42—26no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—43no—
241C2×C42—43no—
242#95—26no—
243#95—26no—
244#95—26no—
245#95—26no—
246#95—26no—
247#95—26no—
248#95—26no—
249#95—43no—
250#95—41no—
251#95—41no—
252#95—43no—
253#95—43no—
254#95—43no—
255#95—43no—
256#95—43no—
257#95—43no—
258#95—43no—
259#95—43no—
260#95—43no—
261#95—43no—
262#95—43no—
263#95—43no—
264#95—43no—
265#95—43no—
266#95—43no—
267#95—43no—
268#95—43no—
269#95—43no—
270#95—43no—
271#95—43no—
272C24—23no—
273C24—23no—
274C24—23no—
275C24—23no—
276C24—23no—
277C24—23no—
278C24—23no—
279C24—23no—
280C24—23no—
281C24—23no—
282C24—23no—
283C24—23no—
284C24—224no—
285C24—412no—
286C24—412no—
287C24—412no—
288C24—412no—
289C24—412no—
290C24—412no—
291C24—412no—
292C24—412no—
293C24—412no—
294C24—44no—
295C24—44no—
296C24—412no—
297C24C2×#4116yes—
298C24H(5,𝔽2)16yes—
299C242−1+416yes—
300C24C2511yescharacteristic
301C2×Q8—86no—
302C2×Q8—86no—
303C2×Q8—82no—
304C22×C4—23no—
305C22×C4—23no—
306C22×C4—23no—
307C22×C4—23no—
308C22×C4—23no—
309C22×C4—23no—
310C22×C4—23no—
311C22×C4—23no—
312C22×C4—23no—
313C22×C4—23no—
314C22×C4—23no—
315C22×C4—23no—
316C22×C4—23no—
317C22×C4—23no—
318C22×C4—23no—
319C22×C4—23no—
320C22×C4—23no—
321C22×C4—23no—
322C22×C4—23no—
323C22×C4—23no—
324C22×C4—23no—
325C22×C4—23no—
326C22×C4—23no—
327C22×C4—23no—
328C22×C4—23no—
329C22×C4—23no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—23no—
339C22×C4—23no—
340C22×C4—23no—
341C22×C4—23no—
342C22×C4—23no—
343C22×C4—23no—
344C22×C4—23no—
345C22×C4—23no—
346C22×C4—23no—
347C22×C4—23no—
348C22×C4—23no—
349C22×C4—23no—
350C22×C4—23no—
351C22×C4—23no—
352C22×C4—23no—
353C22×C4—23no—
354C22×C4—23no—
355C22×C4—23no—
356C22×C4—23no—
357C22×C4—23no—
358C22×C4—23no—
359C22×C4—23no—
360C22×C4—23no—
361C22×C4—23no—
362C22×C4—23no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—23no—
366C22×C4—412no—
367C22×C4—412no—
368C22×C4—412no—
369C22×C4—412no—
370C22×C4—412no—
371C22×C4—412no—
372C22×C4—412no—
373C22×C4—412no—
374C22×C4—412no—
375C22×C4—412no—
376C22×C4—412no—
377C22×C4—412no—
378C22×C4—4128no—
379C22×C4—412no—
380C22×C4—412no—
381C22×C4C2×#4116yes—
382C22×C4C2×#4116yes—
383C22×C4H(5,𝔽2)16yes—
384C22×C4H(5,𝔽2)12yes—
385C4⋊C4—412no—
386C4⋊C4—86no—
387C4⋊C4—86no—
388C4⋊C4—86no—
389C4⋊C4—86no—
390C4⋊C4—86no—
391C4⋊C4—86no—
392C22⋊C4—412no—
393C22⋊C4—412no—
394C22⋊C4—412no—
395C22⋊C4—412no—
396C22⋊C4—412no—
397C22⋊C4—412no—
398C22⋊C4—412no—
399C22⋊C4—86no—
400C22⋊C4—86no—
401C22⋊C4—86no—
402C22⋊C4—86no—
403C22⋊C4—86no—
404C42—44no—
405C42—86no—
406C23—26no—
407C23—26no—
408C23—26no—
409C23—26no—
410C23—26no—
411C23—26no—
412C23—26no—
413C23—26no—
414C23—26no—
415C23—26no—
416C23—26no—
417C23—26no—
418C23—26no—
419C23—26no—
420C23—26no—
421C23—26no—
422C23—26no—
423C23—26no—
424C23—448no—
425C23—448no—
426C23—448no—
427C23—416no—
428C23—448no—
429C23—416no—
430C23—416no—
431C23—43no—
432C23—43no—
433C23—43no—
434C23—43no—
435C23—43no—
436C23—43no—
437C23—43no—
438C23—43no—
439C23—43no—
440C23—43no—
441C23—43no—
442C23—43no—
443C23—43no—
444C23—43no—
445C23—43no—
446C23—43no—
447C23#542112yes—
448C23C22×#4113yes—
449C23C2×H(5,𝔽2)13yes—
450C23C2×H(5,𝔽2)13yes—
451C23C2×H(5,𝔽2)11yescharacteristic
452C23C2×2−1+413yes—
453C23#58511yescharacteristic
454C23#58511yescharacteristic
455Q8—88no—
456C2×C4—26no—
457C2×C4—26no—
458C2×C4—26no—
459C2×C4—26no—
460C2×C4—26no—
461C2×C4—26no—
462C2×C4—26no—
463C2×C4—26no—
464C2×C4—26no—
465C2×C4—26no—
466C2×C4—448no—
467C2×C4—448no—
468C2×C4—43no—
469C2×C4—43no—
470C2×C4—43no—
471C2×C4—43no—
472C2×C4—43no—
473C2×C4—43no—
474C2×C4—43no—
475C2×C4—43no—
476C2×C4—41no—
477C2×C4—43no—
478C2×C4—43no—
479C2×C4—43no—
480C2×C4—43no—
481C2×C4—43no—
482C2×C4—43no—
483C2×C4—43no—
484C2×C4—43no—
485C2×C4—43no—
486C2×C4—43no—
487C2×C4—43no—
488C2×C4—43no—
489C2×C4—43no—
490C2×C4—43no—
491C2×C4—43no—
492C2×C4—43no—
493C2×C4—43no—
494C2×C4—43no—
495C2×C4—43no—
496C2×C4—43no—
497C2×C4—43no—
498C2×C4—43no—
499C2×C4—43no—
500C2×C4—43no—
501C2×C4—43no—
502C2×C4—43no—
503C2×C4—43no—
504C2×C4—43no—
505C2×C4—43no—
506C2×C4—43no—
507C2×C4—43no—
508C2×C4—43no—
509C2×C4—43no—
510C2×C4—43no—
511C2×C4—43no—
512C2×C4—43no—
513C2×C4—43no—
514C2×C4—43no—
515C2×C4—43no—
516C2×C4—43no—
517C2×C4—43no—
518C2×C4—43no—
519C2×C4—43no—
520C2×C4—43no—
521C2×C4—43no—
522C2×C4—43no—
523C2×C4—43no—
524C22—212no—
525C22—212no—
526C22—212no—
527C22—464no—
528C22—46no—
529C22—46no—
530C22—46no—
531C22—46no—
532C22—46no—
533C22—46no—
534C22—46no—
535C22—46no—
536C22—46no—
537C22—46no—
538C22—46no—
539C22—46no—
540C22C2×#54213yes—
541C22#348113yes—
542C22#349213yes—
543C22#350513yes—
544C22#351313yes—
545C22#352013yes—
546C22#352911yescharacteristic
547C22#353213yes—
548C22#353211yescharacteristic
549C22#353213yes—
550C22#353513yes—
551C22#353813yes—
552C22#354113yes—
553C4—46no—
554C4—46no—
555C4—42no—
556C4—46no—
557C4—83no—
558C4—83no—
559C4—83no—
560C4—83no—
561C4—83no—
562C4—83no—
563C2—412no—
564C2#3677813yes—
565C2#3851113yes—
566C2#3853513yes—
567C2#4561413yes—
568C2#5316811yescharacteristic
569C2#5316911yescharacteristic
570C2#6003211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #16429C2semidirect
2. #16318C2semidirect
3. #16005C2semidirect
4. #15823C2semidirect
5. #15622C2semidirect
6. #15544C2semidirect
10. #14547C2semidirect
11. #14433C2semidirect
15. C2×#395C22semidirect
16. C2×#395C22semidirect
19. C2×#387C22semidirect
26. C2×#383C22semidirect
29. #1445C22semidirect
30. #1445C22semidirect
31. #1445C22semidirect
32. #1444C22semidirect
33. #1443C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
57. #1436C22semidirect
58. C4⋊#95C22semidirect
59. C4⋊#95C22semidirect
60. #1434C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#164292
#163182
#160052
#158232
#156222
#155442
#154672
#152972
#146932
#145472
#144332